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18.104
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Spring 2013
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\begin{center}\textbf{Homework 1}\end{center}

\begin{center}Due February 12th by the beginning of class.\end{center}

\textbf{Problem:} Let $u  \colon C_c^\infty((0,\infty)) \to \mathbb C$ be defined by
\[
\langle u, \phi \rangle = \int_0^\infty e^{1/x}\phi(x)dx.
\]
\begin{enumerate}
\item Prove that $u \in \mathscr D '((0,\infty))$
\item Prove that there is no $v \in \mathscr D '(\mathbb R)$ whose restriction to $(0,\infty)$ is $u$. 
\end{enumerate}
\textit{Hint:} The first part has a very short solution. For the second, show that the estimate \cite[(1.3.1)]{frjo} in the definition of a distribution leads to a contradiction by considering a suitable family of test functions $\phi_\epsilon$, parametrized by $\epsilon \in (0,1)$. One possible choice is $\phi_\epsilon(x) = \phi(x/\epsilon)$, for some $\phi \in C_c^\infty((0,\infty))$ nonnegative and not identically zero.

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\textbf{Solution:}


\begin{thebibliography}{00}
\bibitem[FrJo]{frjo} G. Friedlander and M. Joshi. The Theory of Distributions, second edition, Cambridge University Press, 1998.
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